# Full text of "History Of The Theory Of Numbers - I"

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HlSTOKT OF THE THEORY OF NUMBERS.
[CHAP. DC
R. D. Carmichael121 proved that, if a+l and 2a+l are both primes, (a!)4— 1 is divisible by (a+l)(2a+l), and conversely.
A. Are*valo136 proved (6) and Lucas'77 residues of binomial coefficients. N. G. W. H. Beeger137 proved that [if p is a prime]
(p-l)!+ls«-p+l (mod?2), 5 = 1+2^+. . .+(p-ir"1-A-i, where h is a Bernoulli number defined by the symbolical equation (/i+l)n =/in, fci=l/2. By use of Adams'137" table of hi, K114, it was verified that p = 5, p = 13 are the only p<114 for which (p — 1)1+1=0 (mod p2).
T. E. Mason138 and J. M. Child139 noted that, if p is a prime > 3,
(np)l=nl(pl)n (modptt+3).
N. Nielsen140 proved that, if p = 2n+l, P=l-3-5. . . (2n-l), P2s= ( - I)n22n(2n) !   (mod p2),
If p is a prime >3, P=(-l)n23nn!   (mod p3).    He gave the last result also elsewhere.141
C. I. Marks142 found the smallest integer x such that 24 ... (2n)x is divisible by 3-5. .. (2n-l).
186Revista de la Sociedad Mat. Espafiola, 2,
1913,130-1.
"'Messenger Math., 43, 1913-4, 83HL "7«Jour. fiir Math., 85, 1878, 269-72. U8T6hoku Math. Jour., 5,1914, 137.
U9Math. Quest. Educat. Times, 26, 1914, 19.
"°AjinaIi di mat., (3), 22, 1914, 81-2.
lttK. Danske Vidensk. Selsk. Skrifter, (7), 10
1913, 353. "*Math. Quest. Educ. Times, 21, 1912, 84-6.```